The bearings were manufactured according to the commonly used ISO-492-2002 standard. This international standard specifies tolerances for the limiting dimensions and running accuracy of radial roller bearings specified in ISO 15, ISO 355, and ISO 8443.

The requirements for the quality of the final surface depend mainly on the material from which the bearings are made. In particular, it is about the possible surface defects and the depth of the decarburised layer. The surface of the rolled tubes must be free from surface defects such as scratches and grooves, cracks or scales and must not be burnt or heated (etching tests are carried out after grinding to identify burnt or heated material). If these defects were to occur, they would have to be removed uneconomically by machining an increased material allowance or discarded outright as non-conforming pieces.

Decarburisation (up to a certain depth of material) occurs during manufacturing operations (heating, rolling, soft annealing) in which a protective atmosphere is not provided. Quality removal of the decarburised layer is only achieved by chip machining (turning, grinding).

Other specific requirements for bearing ring surfaces are their roughness, roundness, and waviness.

Lower surface roughness is desirable on orbits because it causes less wear, greater durability, and lower noise. While the roughness on non-functional surfaces is Ra = 0.5 µm, on functional surfaces (loaded by rolling element circulation) a lower roughness is required as standard, Ra = 0.15 µm or Ra = 0.12 µm.

The roundness of the functional and non-functional surfaces of the bearing rings is specified according to the standard (ČSN ISO 1132). The prescribed roundness on the functional raceway surfaces significantly influences the bearing running quality (inner diameter at the outer ring, outer diameter at the inner ring, and at the outer diameter of the rolling element). From the point of view of correct bearing mounting, non-functional roundness (on the outer diameter of the outer ring and on the inner diameter of the inner ring) is allowed to be up to an order of magnitude greater, as these surfaces do not affect the bearing running. The roundness of the orbit can be viewed as a profile of true shape moving around an ideal circle in positive and negative deviations oscillating regularly in “waves”. In practice, low wave count deviations are evaluated when the number of waves is less than 14 per revolution. For the presented bearing rings, the roundness is in the order of um, or a maximum deviation of 2 um per functional area.

Waviness, like roundness, is evaluated in relation to an ideal circle, with the number of deviating “waves” typically greater than 14 per revolution. The waviness is particularly affected by finishing operations (grinding, superfinishing). The waviness is assessed from the same recorded signal as the roundness, filtering out only the frequencies included in the roundness. The corrugation of the presented bearing rings was given by the internal standard of JTEKT Bearing in Olomouc (after renaming the former KOYO company), which is specified as different from the corrugations of rolling elements to avoid undesirable resonance phenomena.

Stabilizing the dimensional accuracy of bearing rings means, in particular, stabilizing their manufacturing process. It is necessary to have real-time information about the manufacturing process based on data obtained from sub-processes and to work with “post-process” gauges, i.e., gauges designed for the output of sub-operations. This data is crucial to send a signal to the machine tool (lathe or grinder) to make corrections. In the case of turning and grinding, post-process gauges can be effectively used to measure geometric dimensions and geometric tolerances quickly. Grinding is a more efficient manufacturing operation than turning, and cycle times are shorter. Therefore, there is no time for measuring pieces at the output in engineering practice. However, modern grinding machines have electronic speed control, they are very reliable and precise production machines that routinely achieve high accuracy (up to 0.001 mm) and generate surface roughness up to 0.025 μm when grinding. When the requirement for regular maintenance and adjustment by trained operators is met, the demands for dimensional accuracy can be fully accepted.

Heat treatment (quenching) is such a variable process that stabilization of dimensional accuracy is not assured by, for example, metallographic tests, cut-outs or hardness measurements for each hardened production batch. Again, this must be based on the collection and interpretation of data that can be used to predict how well a given batch will be hardened. A factor significantly affecting the production process is the batch of material, which may vary slightly in chemical composition. This is where the measurement of internal stresses in the material by non-destructive testing (X-ray or ultrasonic) is helpful. Tempering then reduces the stresses and brittleness of the steel after pre-hardening and ensures dimensional stabilisation.

Bearings are an essential component of various machines, and the precision of their dimensions and running accuracy is of paramount importance when it comes to their production and application. Stabilizing the dimensional accuracy of bearing rings means, in particular, stabilizing their manufacturing process. It is necessary to have real-time information about the manufacturing process based on data obtained from sub-processes and to work with “post-process” gauges, i.e., gauges designed for the output of sub-operations. This data is crucial to send a signal to the machine tool (lathe or grinder) to make corrections. In the case of turning and grinding, post-process gauges can be effectively used to measure geometric dimensions and geometric tolerances quickly. Grinding is a more efficient manufacturing operation than turning, and cycle times are shorter. Therefore, there is no time for measuring pieces at the output in engineering practice. However, modern grinding machines have electronic speed control, they are very reliable and precise production machines that routinely achieve high accuracy (up to 0.001 mm) and generate surface roughness up to 0.025 μm when grinding. When the requirement for regular maintenance and adjustment by trained operators is met, the demands for dimensional accuracy can be fully accepted.

Heat treatment (quenching) is such a variable process that stabilization of dimensional accuracy is not assured by, for example, metallographic tests, cut-outs or hardness measurements for each hardened production batch. Again, this must be based on the collection and interpretation of data that can be used to predict how well a given batch will be hardened. A factor significantly affecting the production process is the batch of material, which may vary slightly in chemical composition. This is where the measurement of internal stresses in the material by non-destructive testing (X-ray or ultrasonic) is helpful. Tempering then reduces the stresses and brittleness of the steel after pre-hardening and ensures dimensional stabilisation.

The manufacturing quality of bearing rings in terms of their surface roughness, roundness, parallelism, and various geometric tolerances is now very well studied. The presented publication has new objectives: to offer an alternative to destructive testing of bearing rings, to pave the way for the prediction of residual stresses in materials, to communicate clearly and unambiguously with suppliers of semi-finished products (especially tubes and rods) regarding material quality. In fact, standard metallographic tests (especially sections, hardness measurements, etc.) are laboratory tests with no predictive value about the internal stresses in the material. Internal stresses are present in all manufacturing operations because each machining operation introduces more and more stresses into the material, starting with turning, hardening, tempering, grinding, superfinishing and assembly. It can, therefore, be assumed that internal stresses have a primary effect on the resulting bearing life, if extreme, even a well machined surface finish will not increase the life of such a bearing ring to the desired level.

The material used was 100Cr6 steel type 1.3505 (Table 1), which is one of the most commonly used grades of bearing steels from the group of quenchable bearing steels. In terms of chemical composition, it is a high-carbon, low-alloy steel with chromium addition, mainly used in the production of rolling bearings. Its dominant mechanical properties are high hardness, wear resistance, and long service life. As this steel achieves Rc values of 60–67 at room temperature (Rockwell hardness), it has, in particular, high resistance to pressure and abrasion.

Due to the excellent physical and mechanical properties of this steel, its applications in technical practice are wide; they are used mainly in abrasion-stressed automotive components (brakes, steering, line shaft), as well as in agitators, sliders, quick couplings, machine tools, locking mechanisms, conveyor belts, skates, springs, pumps, measuring instruments, valves, etc.

The basic principle of the X-ray diffraction method is the interaction of X-rays with the crystal lattice. The scattering of X-rays on crystals shows the changes in the distances of the atomic lattice planes caused by the residual stress of the component. The interaction of the X-ray beam with the crystal lattice of the material under investigation occurs under the interference of scattered X-rays, according to Bragg’s law. The deformations that are recorded by this method are then converted by elasticity theory into stresses using special computational methods with the necessary elastic constants [x].

The Proto iXRD mobile diffractometer (Figure 1), which operates on the principle of a non-destructive method, was selected to measure the residual stresses on the bearing steel. Due to the positioning of the detector away from the sensing head and fibre optics, it allows the measurement of rough surfaces in very small spaces and also in curved areas that are more difficult to reach. It uses three X-ray anodes (Mn, Cr, Cu), which can measure virtually all materials.

Due to the properties of the measured material, a Cr Kα lamp was used. The bearings were thoroughly cleaned and placed on an automated adjustable table under the measuring head, with the measured bearing surfaces parallel to the axis of the diffractometer arm. Prior to the actual measurement, the distance between the collimator and the measurement area was adjusted using the mounted Pointer on the collimator. Subsequently, the Auto Focus command was run, and the structure of the material to be measured was set to martensite in the Constants window of the measurement software.

The next step in the workflow was to place a titanium plate on the measurement surface to filter out interfering elements. After running the Gain command, an exposure displaying the profile began to form. After the profile was displayed, the plate was removed from the measurement surface and point (axial) and triaxial residual stress measurements followed.

A collimator with a diameter of 2 mm was used to focus the X-rays during the measurements, and the exposure time was set to 2 s to achieve the optimum intensity of the diffraction peak for tilting the measuring head. In the case of the triaxial residual stress measurements, the measurements were carried out while the sample was rotated on its axis, specifically by 60°, i.e., a total of 6 measurements were made in one rotation cycle. After the measurements, the Xrd Win measurement and evaluation software generated the measurement results and the corresponding result reports (Valíček et al., 2019; Šajgalík et al., 2018).

Surface roughness measurements were performed with the Alicona InfiniteFocus G5plus, which is a highly accurate, fast and flexible 3D optical device combining dimensional metrology and surface roughness measurement in one system. The measurement results are repeatable, with a possible vertical resolution of up to 10 nm. The reliable Focus Variation measurement combined with a high-quality anti-vibration system enables shape and roughness measurements also for large and heavy components directly in the production process. Infinite Focus is equipped with high-precision encoders, ensuring precise movement in all axes. With the automation interface, Infinite Focus can also be used for fully automatic measurements.

All measurements of the residual stresses of bearing steels after technological treatments were performed with the Proto iXRD diffractometer in the laboratory of the Department of Machining and Manufacturing Technologies at the Faculty of Mechanical Engineering, University of Žilina (Figure 3).

Figure 3 shows the measurement of the residual stress σ _nap of turned-bearing steel L _5tur in the centre of its cylindrical surface at the angular position ψ = 0° by a Proto iXRD diffractometer. An analogous procedure was followed to measure the residual stresses in the other five angular positions (60°, 120°, 180°, 240°, 300°). Also, the measurement of the residual stresses σ _rez of the bearing steel L _xque after quenching was carried out using the same workflow.

The measurement results presented in the form of regression relationships are, therefore, based on the setting of specific technological parameters, specific parameters and measurement conditions and the specific material used. For different settings, the regression relationships will only have a similar predictive value. It can be concluded that residual stresses of significantly loaded materials significantly affect the functionality of bearings. Therefore, their identification must not be neglected in engineering practice.

A total of 8 different bearings were measured, 4 of which were in the quenched and tempered condition, and 4 were turned after heat treatment. All bearings were measured at the centre of their cylindrical surface, at six angular positions ψ (Figure 3). The results of the measurements were continuously recorded in tables and graphs, and the corresponding relationships predicting the residual stresses were assigned to them using an approximation method. The values of the residual stress measurements at the corresponding angular position ψ were interleaved with polynomial curves of degree 3 described by the corresponding regression equation with total accuracy of measurement (R ² = 1 in all cases investigated).

The selected angular positions corresponding to the presented tables, figures and equations: ψ = 0°, Table 2 and Figure 4, (Eqs 3, 4); ψ = 60°, Table 3 and Figure 5, (Eqs 5, 6); ψ = 120°, Table. 4 and Figure 6, (Eqs 7, 8); ψ = 180° and Table 5, Figure 7, (Eqs 9, 10); ψ = 240°, Table 6 and Figure 8, (Eqs 11, 12); ψ = 300°, Table 7 and Figure 9, (Eqs 13, 14).

Tables 2–7 contain 5 columns, with columns 1 and 2 presenting the L _x deposits according to their state of processing. Column 3 contains the values of the measured residual stresses σ _rez at the angular positions ψ.

From Figure 4, we got Eqs 3, 4 ψ = 0 ° ;   σ r e z = 111.6 ⋅ σ x t u r 3 − 758.3 ⋅ σ x t u r 2 + 1490.1 ⋅ σ x t u r − 261.3 (3) ψ = 0 ° ;   σ r e z = − 22.3 ⋅ σ x q u e 3 + 255.2 ⋅ σ x q u e 2 − 879.5 ⋅ σ x q u e + 972.8 (4)

From Figure 5, we got Eqs 5, 6 ψ = 60 ° ;   σ r e z = 87.7 ⋅ σ x t u r 3 − 566.8 ⋅ σ x t u r 2 + 1030.6 ⋅ σ x t u r + 67.1 (5) ψ = 0 ° ;   σ r e z = − 6.3 ⋅ σ x q u e 3 + 51.8 ⋅ σ x q u e 2 − 128.5 ⋅ σ x q u e + 144.6 (6)

From Figure 6, we got Eqs 7, 8 ψ = 120 ° ;   σ r e z = 107.95 ⋅ σ x t u r 3 − 726.8 ⋅ σ x t u r 2 + 1398.7 ⋅ σ x s ú s − 136.4 (7) ψ = 120 ° ;   σ r e z = − 24.5 ⋅ σ x q u e 3 + 180.5 ⋅ σ x q u e 2 − 378.7 ⋅ σ x q u e + 238.2 (8)

From Figure 7, we got Eqs 9, 10 ψ = 180 ° ;   σ r e z = 113.1 ⋅ σ x t u r 3 − 750.2 ⋅ σ x t u r 2 + 1407.8 ⋅ σ x t u r − 125.5 (9) ψ = 180 ° ;   σ r e z = − 61.7 ⋅ σ x q u e 3 + 430.2 ⋅ σ x q u e 2 − 866.9 ⋅ σ x q u e + 570.5 (10)

From Figure 8, we got Eqs 11, 12 ψ = 240 ° ;   σ r e z = 148.4 ⋅ σ x t u r 3 − 1024.1 ⋅ σ x t u r 2 + 2054.4 ⋅ σ x t u r − 588.3 (11) ψ = 240 ° ;   σ r e z = 25.7 ⋅ σ x q u e 3 − 199.5 ⋅ σ x q u e 2 + 422.3 ⋅ σ x q u e − 158.6 (12)

From Figure 9, we got Eqs 13, 14 ψ = 300 ° ;   σ r e z = 67.5 ⋅ σ x t u r 3 − 433.9 ⋅ σ x t u r 2 + 787.5 ⋅ σ x t u r + 200.5 (13) ψ = 300 ° ;   σ r e z = 13.6 ⋅ σ x q u e 3 − 83.4 ⋅ σ x q u e 2 + 70.4 ⋅ σ x q u e + 208.0 (14)

The results of the measurements recorded in the tables (Tables 2–7) were the basis for the construction of the graph (Figure 10) expressing the comparative waveforms of the residual stresses σ _rez for L _xque bearings after heat treatment, depending on the selected angular positions ψ. The waveforms were assigned the corresponding relations predicting the residual stresses using an approximation method. The values of the residual stress measurements at the corresponding angular position ψ were interleaved with the polynomial curves of degree 5 described by the corresponding regression equation with total accuracy of measurement (R ² = 1 in all investigated cases) (Eqs 15–18). σ r e z = 2 ⋅ 10 − 9 ⋅ σ 1 q u e 5 − 1 ⋅ 10 − 6 ⋅ σ 1 q u e 4 + 0.000 ⋅ σ 1 q u e 3 + 0.022 ⋅ σ 1 q u e 2 − 6.171 ⋅ σ 1 q u e + 326.1 (15) σ r e z = 2 ⋅ 10 − 9 ⋅ σ 2 q u e 5 − 2 ⋅ 10 − 6 ⋅ σ 2 q u e 4 + 0.000 ⋅ σ 2 q u e 3 − 0.068 ⋅ σ 2 q u e 2 + 2.167 ⋅ σ 2 q u e + 55.8 (16) σ r e z = 1 ⋅ 10 − 8 ⋅ σ 3 q u e 5 − 8 ⋅ 10 − 6 ⋅ σ 3 q u e 4 + 0.002 ⋅ σ 3 q u e 3 − 0.197 ⋅ σ 3 q u e 2 + 6.593 ⋅ σ 3 q u e + 28.0 (17) σ r e z = − 2 ⋅ 10 − 9 ⋅ σ 4 q u e 5 + 1 ⋅ 10 − 6 ⋅ σ 4 q u e 4 − 0.000 L ⋅ σ 4 q u e 3 + 0.035 ⋅ σ 4 q u e 2 − 2.184 ⋅ σ 4 q u e + 108.8 (18)

The results of the measurements recorded in the tables (Tables 2–7) were the basis for the construction of the graph (Figure 11) expressing the comparative waveforms of the residual stresses σ _rez of the L _xtur bearings after quenching and subsequent turning, depending on the selected angular positions ψ. The waveforms were assigned the corresponding relations predicting the residual stresses using an approximation method. The values of the residual stress measurement results at the respective angular position ψ were interleaved with the polynomial curves of degree 5 described by the respective regression equation with total accuracy of measurement (R ² = 1 in all investigated cases) (Eqs 19–22). σ r e z = 2 ⋅ 10 − 9 ⋅ σ 1 t u r 5 − 1 ⋅ 10 − 6 ⋅ σ 1 t u r 4 + 0.000 ⋅ σ 1 t u r 3 − 0.025 ⋅ σ 1 t u r 2 + 1.397 ⋅ σ 1 t u r + 582.0 (19) σ r e z = − 5 ⋅ 10 − 9 ⋅ σ 2 t u r 5 + 4 ⋅ 10 − 6 ⋅ σ 2 t u r 4 − 0.001 ⋅ σ 2 t u r 3 + 0.118 ⋅ σ 2 t u r 2 − 4.329 ⋅ σ 2 t u r + 578.4 (20) σ r e z = 7 ⋅ 10 − 10 ⋅ σ 3 t u r 5 − 3 ⋅ 10 − 7 ⋅ σ 3 t u r 4 + 2 ⋅ 10 − 5 ⋅ σ 3 t u r 3 − 0.002 ⋅ σ 3 t u r 2 + 0.599 ⋅ σ 3 t u r + 397.0 (21) σ r e z = 4 ⋅ 10 − 10 ⋅ σ 4 t u r 5 − 3 ⋅ 10 − 7 ⋅ σ 4 t u r 4 + 0.000 ⋅ σ 4 t u r 3 − 0.017 ⋅ σ 4 t u r 2 + 1.108 ⋅ σ 4 t u r + 707.3 (22)

The mean values of the residual stresses of L _x bearings at the angular positions ψ were evaluated as arithmetic averages, namely, as the mean values of a for L _xque bearings after heat treatment (Eq. 23) and as the mean value of b for the L _xtur bearings after heat treatment and subsequent turning (Eq. 24). a = σ 1 q u e + σ 2 q u e + σ 3 q u e + σ 4 q u e 4 (23) b = σ 1 t u r + σ 2 t u r + σ 3 t u r + σ 4 t u r 4 (24)

The course of these mean values in angular positions is shown in the graph in Figure 12.

The values of the residual stress results measured on L _xque and L _xtur bearings at the respective angular positions were interleaved with the polynomial curves of degree 5 (Figure 12) described by the respective regression equations with total accuracy of measurement (in all investigated cases R ² = 1) (Eqs 25, 26). σ r e z = 3 ⋅ 10 − 9 ⋅ σ x q u e 5 − 3 ⋅ 10 − 6 ⋅ σ x q u e 4 + 6 ⋅ 10 − 4 ⋅ σ x q u e 3 − 0.052 ⋅ σ x q u e 2 + 0.101 ⋅ σ x q u e + 129.68 (25) σ r e z = − 5 ⋅ 10 − 10 ⋅ σ x t u r 5 + 5 ⋅ 10 − 7 ⋅ σ x t u r 4 − 2 ⋅ 10 − 4 ⋅ σ x t u r 3 + 0.018 ⋅ σ x t u r 2 − 0.306 ⋅ σ x t u r + 566.20 (26)

Figures 13, 14 show how uniformly the residual stresses of the bearing steel are distributed after the technological treatments, i.e., how the measured values trend or oscillate around the mean value and what the interval of dispersion is.

The mean value of the residual stress on the thermally treated bearings measured at all angular positions is 82.35 MPa. The variance of the residual stresses measured at the selected angular positions increases and then decreases (Figure 13). Considering that the variance of the investigated deviations is measurable in the total interval of 97.73 MPa, it can be concluded that it is relatively significant with respect to the overall mean value of the residual stresses.

The mean value of the residual stress on the heat treated and subsequently turned bearings measured at all selected angular positions is 588.23 MPa. The variance of the residual stresses measured at the selected angular positions is alternately decreasing, increasing and decreasing (Figure 14). Considering the fact that the variance of the investigated variance is measurable over the total interval of 1,151.12 MPa, it can be concluded that it is relatively significant with respect to the determined mean value of the residual stresses and higher than the residual stress distribution on the heat-treated bearing steel only.

In the case of surface roughness measurements, the surfaces of quenched and turned bearings were measured, with the measurements repeated ten times. The roughness parameters Ra, Rq, Rz were measured comparatively, with Ra chosen as the most relevant parameter for the comparison of values. Measurements were again taken at each of the bearing positions. It can be pointed out that when comparing the measured roughness values (Tables 8, 9; Figure 15), the values of quenched bearings are much higher than those of turned bearings. However, no relatively significant deviation from the standard is evident for either group of measurements.

The initial basis for deriving further analytical relations in this application case is also the copyrighted patent, from which relations (Eqs 27–29) were taken for further analytical description. σ r e z = σ r z ⋅ cos δ ⋅ π 180 2 (27) σ r e z 0 = σ r z 0 ⋅ cos δ 0 ⋅ π 180 2 (28) or σ r e z 0 = R p 0.2 ⋅ cos δ 0 ⋅ π 180 2 (29)

After adjustment, relationships involving the independent variable Ra (Eqs 30–32) can be obtained, σ r e z = 10 − 3 ⋅ R a ⋅ E m a t R a 0 ⋅ cos δ ⋅ π 180 2 (30) from which we can obtain the relation (Eq. 31) for the roughness Ra, where Ra ₀ roughness on the neutral plane h ₀ is equal to 3.7 µm = const. R a = 10 3 ⋅ σ r e z ⋅ R a 0 E m a t ⋅ cos δ ⋅ π 180 2 (31)

It can be assumed that the source of residual stress generation is the mechanical condition of the surface layer, which is reflected in the surface roughness Ra, especially in the cutting tool marks. The authors’ new ambition was to analytically capture the influence of the subsurface layers down to the core of the material (Harničárová et al., 2013; Valíček et al., 2020). In this respect, the structural grain size D _gr and its alteration due to stresses of different nature also gained importance. On this basis, relationships (Eqs 32–39) can be derived. In these new relations, the broader dependence σ _rez = f (D _gr , Ra) is already respected. Theoretically, other related dependent variables resulting from, e.g., the graph in Figure 16 can be added to this implicit equation. We know that capturing a larger number of dependent variables analytically makes the calculations more accurate and increases their predictive and application capabilities.

After adjustment, we obtain relationships involving both independent variables, i.e., both surface roughness and structural grain size, according to relations (Eqs 32–39). R a r e z S = 10 2 ⋅ R a 0 ⋅ R a 0 ⋅ D g r 0 0.6419 ⋅ σ r e z S cos K p l m a t ⋅ π 180 2 ⋅ E m a t (32) R a r e z K = 10 2 ⋅ R a 0 ⋅ R a 0 ⋅ D g r 0 0.6419 ⋅ σ r e z K cos K p l m a t ⋅ π 180 2 ⋅ E m a t (33) σ r e z S = 10 − 3 ⋅ R a r e z S ⋅ E m a t R a 0 (34) σ r e z K = 10 − 3 ⋅ R a r e z K ⋅ E m a t R a 0 (35)

The following relations (Eqs 36–39) apply analogously for the structural grain size. D g r S = 10 3 ⋅ R a 0 ⋅ σ r e z S E m a t (36) D g r K = 10 3 ⋅ R a 0 ⋅ σ r e z K E m a t (37) respectively σ r e z S = 10 − 3 ⋅ D g r S ⋅ E m a t R a 0 (38) σ r e z K = 10 − 3 ⋅ D g r K ⋅ E m a t R a 0 (39)

Symbols used in Eqs 27–39: